Question

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was 5/7, and therefore the probability that the Democratic candidate would be elected was 2/7 (the two Independent candidates were given no chance of being elected). It was also estimated that if the Republican candidate were elected, the probability that a conservative, moderate, or liberal judge would be appointed to the Supreme Court (one retirement was expected during the presidential term) was 1/7, 1/7, and 5/7, respectively. If the Democratic candidate were elected, the probabilities that a conservative, moderate, or liberal judge would be appointed to the Supreme Court would be 1/3, 1/6, and 1/2, respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected

Answer 1

Answer:

The probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is $\frac{14}{29}$.

Step-by-step explanation:

The conditional probability of an events A given that another events B has already occurred is given by:

$P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^{c})P(A^{c})}$

The estimation of the past presidential election is provided.

Denote the events as follows:

R = a Republican candidate would be elected

D = a Democratic candidate would be elected

C = a conservative judge would be appointed to the Supreme Court

M = a moderate judge would be appointed to the Supreme Court

L = a liberal judge would be appointed to the Supreme Court

The information provided is:

$P(R)=\frac{5}{7},\ P(D)=\frac{2}{7}$

$P(C|R)=\frac{1}{7},\ P(M|R)=\frac{1}{7},\ P(L|R)=\frac{5}{7}$

$P(C|B)=\frac{1}{3},\ P(M|D)=\frac{1}{6},\ P(L|D)=\frac{1}{2}$

It is provided that a conservative judge was appointed to the Supreme Court during the presidential term.

Compute the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court as follows:

$P(D|C)=\frac{P(C|D)P(D)}{P(C|D)P(D)+P(C|R)P(R)}$

             $=\frac{(\frac{1}{3}\times \frac{2}{7})}{(\frac{1}{3}\times \frac{2}{7})+(\frac{1}{7}\times \frac{5}{7})}$

             $=\frac{\frac{2}{21}}{\frac{2}{21}+\frac{5}{49}}$

             $=\frac{2}{21}\div \frac{29}{147}$

             $=\frac{14}{29}$

Thus, the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is $\frac{14}{29}$.